X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fcomputation%2Flpxs_lleq.ma;h=42e34097a4db92ba9d9e7bf4c3375f679b41d747;hb=33f8507cadd3b36dc9afa227d8968dda66fe2034;hp=a78d6c70d289a10b0eecbde72c1444c71357ad8b;hpb=606dab57f31b66eb3f30f603185124b88dfad4c1;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma index a78d6c70d..42e34097a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma @@ -12,110 +12,32 @@ (* *) (**************************************************************************) -include "basic_2/substitution/lleq_ext.ma". -include "basic_2/computation/lpxs_ldrop.ma". +include "basic_2/reduction/lpx_lleq.ma". +include "basic_2/computation/cpxs_leq.ma". +include "basic_2/computation/lpxs_drop.ma". include "basic_2/computation/lpxs_cpxs.ma". (* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************) -(* Advanced properties ******************************************************) - -fact le_repl_sn_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z. -// qed-. - -axiom cpxs_cpys_conf_lpxs: ∀h,g,G,d,e. - ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h, g] T1 → - ∀T2. ⦃G, L0⦄ ⊢ T0 ▶*×[d, e] T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡*[h, g] L1 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ▶*×[d, e] T & ⦃G, L0⦄ ⊢ T2 ➡*[h, g] T. - -axiom cpxs_conf_lpxs_lpys: ∀h,g,G,d,e. - ∀I,L0,V0,T0,T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡*[h, g] T1 → - ∀L1. ⦃G, L0⦄ ⊢ ➡*[h, g] L1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ▶*×[d, e] V2 → - ∃∃T. ⦃G, L1.ⓑ{I}V0⦄ ⊢ T1 ▶*×[⫯d, e] T & ⦃G, L0.ⓑ{I}V2⦄ ⊢ T0 ➡*[h, g] T. - - -include "basic_2/reduction/cpx_cpys.ma". - -fact pippo_aux: ∀h,g,G,L1,T,T1,d,e. ⦃G, L1⦄ ⊢ T ▶*×[d, e] T1 → e = ∞ → - ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → - ∃∃T2. ⦃G, L2⦄ ⊢ T ▶*×[d, e] T2 & ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 & - L1 ⋕[T, d] L2 ↔ T1 = T2. -#h #g #G #L1 #T #T1 #d #e #H @(cpys_ind_alt … H) -G -L1 -T -T1 -d -e [ * ] -[ /5 width=5 by lpxs_fwd_length, lleq_sort, ex3_intro, conj/ -| #i #G #L1 elim (lt_or_ge i (|L1|)) [2: /6 width=6 by lpxs_fwd_length, lleq_free, le_repl_sn_aux, ex3_intro, conj/ ] - #Hi #d elim (ylt_split i d) [ /5 width=5 by lpxs_fwd_length, lleq_skip, ex3_intro, conj/ ] - #Hdi #e #He #L2 elim (lleq_dec (#i) L1 L2 d) [ /4 width=5 by lpxs_fwd_length, ex3_intro, conj/ ] - #HnL12 #HL12 elim (ldrop_O1_lt L1 i) // -Hi #I #K1 #V1 #HLK1 - elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2 - elim (lpxs_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct - elim (lift_total V2 0 (i+1)) #W2 #HVW2 - @(ex3_intro … W2) /2 width=7 by cpxs_delta, cpys_subst/ -I -K1 -V1 -Hdi - @conj #H [ elim HnL12 // | destruct elim (lift_inv_lref2_be … HVW2) // ] -| /5 width=5 by lpxs_fwd_length, lleq_gref, ex3_intro, conj/ -| #I #G #L1 #K1 #V #V1 #T1 #i #d #e #Hdi #Hide #HLK1 #HV1 #HVT1 #_ #He #L2 #HL12 destruct - elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2 - elim (lpxs_inv_pair1 … H) -H #K2 #W #HK12 #HVW #H destruct - elim (cpxs_cpys_conf_lpxs … HVW … HV1 … HK12) -HVW -HV1 -HK12 #W1 #HW1 #VW1 - elim (lift_total W1 0 (i+1)) #U1 #HWU1 - lapply (ldrop_fwd_drop2 … HLK1) -HLK1 #HLK1 - @(ex3_intro … U1) /2 width=10 by cpxs_lift, cpys_subst/ -| #a #I #G #L #V #V1 #T #T1 #d #e #HV1 #_ #IHV1 #IHT1 #He #L2 #HL12 - elim (IHV1 … HL12) // -IHV1 #V2 #HV2 #HV12 * #H1V #H2V - elim (IHT1 … (L2.ⓑ{I}V2)) /4 width=3 by lpxs_cpx_trans, lpxs_pair, cpys_cpx/ -IHT1 -He #T2 #HT2 #HT12 * #H1T #H2T - elim (cpxs_conf_lpxs_lpys … HT12 … HL12 … HV1) -HT12 -HL12 -HV1 #T0 #HT20 #HT10 - @(ex3_intro … (ⓑ{a,I}V2.T0)) - [ @cpys_bind // @(cpys_trans_eq … T2) /3 width=5 by lsuby_cpys_trans, lsuby_succ/ - | /2 width=1 by cpxs_bind/ - | @conj #H destruct - [ elim (lleq_inv_bind … H) -H #HV #HT >H1V -H1V // - | @lleq_bind /2 width=1/ - - - /3 width=5 by lsuby_cpys_trans, lsuby_succ/ -| #I #G #L #V #V1 #T #T1 #d #e #HV1 #HT1 #IHV1 #IHT1 #He #L2 #HL12 - elim (IHV1 … HL12) // -IHV1 #V2 #HV2 #HV12 * #H1V #H2V - elim (IHT1 … HL12) // -IHT1 #T2 #HT2 #HT12 * #H1T #H2T -He -HL12 - @(ex3_intro … (ⓕ{I}V2.T2)) /2 width=1 by cpxs_flat, cpys_flat/ - @conj #H destruct [2: /3 width=1 by lleq_flat/ ] - elim (lleq_inv_flat … H) -H /3 width=1 by eq_f2/ -] - - - - [ - | @cpxs_bind // - @(lpx_cpxs_trans … HT12) -| -] - -axiom lleq_lpxs_trans: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ∀K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 → - ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ⋕[T, d] K2. -(* -#h #g #G #L1 #L2 #T #d #H @(lleq_ind_alt … H) -L1 -L2 -T -d -[ -| -| -| -| -| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #K2 #HLK2 - elim (IHV … HLK2) -IHV #KV #HLKV #HV - elim (IHT (K2.ⓑ{I}V)) -IHT /2 width=1 by lpxs_pair_refl/ -HLK2 #Y #H #HT - elim (lpxs_inv_pair1 … H) -H #KT #VT #HLKT #_ #H destruct - -#h #g #G #L1 #L2 #T #d * #HL12 #IH #K2 #HLK2 -*) - (* Properties on lazy equivalence for local environments ********************) -lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 → - ∃∃K2. ⦃G1, K1, T1⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2. +lemma lleq_lpxs_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 → + ∀L1,T,d. L1 ≡[T, d] L2 → + ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ≡[T, d] K2. +#h #g #G #L2 #K2 #H @(lpxs_ind … H) -K2 /2 width=3 by ex2_intro/ +#K #K2 #_ #HK2 #IH #L1 #T #d #HT elim (IH … HT) -L2 +#L #HL1 #HT elim (lleq_lpx_trans … HK2 … HT) -K +/3 width=3 by lpxs_strap1, ex2_intro/ +qed-. + +lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → + ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 → + ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2. #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 … H1) -H1 #K0 #V0 #H1KL1 #_ #H destruct elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 // - #I1 #K1 #H #H2KL1 lapply (ldrop_inv_O2 … H) -H #H destruct + #K1 #H #H2KL1 lapply (drop_inv_O2 … H) -H #H destruct /2 width=4 by fqu_lref_O, ex3_intro/ | * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H [ elim (lleq_inv_bind … H) @@ -126,20 +48,20 @@ lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, | #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H /2 width=4 by fqu_flat_dx, ex3_intro/ | #G1 #L1 #L #T1 #U1 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1 - elim (ldrop_O1_le (e+1) K1) + elim (drop_O1_le (Ⓕ) (e+1) K1) [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 // - #H2KL elim (lpxs_ldrop_trans_O1 … H1KL1 … HL1) -L1 - #K0 #HK10 #H1KL lapply (ldrop_mono … HK10 … HK1) -HK10 #H destruct + #H2KL elim (lpxs_drop_trans_O1 … H1KL1 … HL1) -L1 + #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct /3 width=4 by fqu_drop, ex3_intro/ - | lapply (ldrop_fwd_length_le2 … HL1) -L -T1 -g + | lapply (drop_fwd_length_le2 … HL1) -L -T1 -g lapply (lleq_fwd_length … H2KL1) // ] ] qed-. -lemma lpxs_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 → - ∃∃K2. ⦃G1, K1, T1⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2. +lemma lpxs_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → + ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 → + ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2. #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1 elim (fquq_inv_gen … H) -H [ #H elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1 @@ -148,9 +70,9 @@ elim (fquq_inv_gen … H) -H ] qed-. -lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 → - ∃∃K2. ⦃G1, K1, T1⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2. +lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → + ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 → + ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2. #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 [ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1 /3 width=4 by fqu_fqup, ex3_intro/ @@ -160,9 +82,9 @@ lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G ] qed-. -lemma lpxs_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 → - ∃∃K2. ⦃G1, K1, T1⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2. +lemma lpxs_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → + ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 → + ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2. #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1 elim (fqus_inv_gen … H) -H [ #H elim (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1 @@ -170,3 +92,33 @@ elim (fqus_inv_gen … H) -H | * #HG #HL #HT destruct /2 width=4 by ex3_intro/ ] qed-. + +fact leq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ⩬[d, e] L0 → e = ∞ → + ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 → + ∃∃L. L ⩬[d, e] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L & + (∀T. L0 ≡[T, d] L2 ↔ L1 ≡[T, d] L). +#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e +[ #d #e #_ #L2 #H >(lpxs_inv_atom1 … H) -H + /3 width=5 by ex3_intro, conj/ +| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct +| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H + elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct + lapply (ysucc_inv_Y_dx … He) -He #He + elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH + @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/ + #T elim (IH T) #HL0dx #HL0sn + @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/ +| #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H + elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct + elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH + @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/ + #T elim (IH T) #HL0dx #HL0sn + @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/ +] +qed-. + +lemma leq_lpxs_trans_lleq: ∀h,g,G,L1,L0,d. L1 ⩬[d, ∞] L0 → + ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 → + ∃∃L. L ⩬[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L & + (∀T. L0 ≡[T, d] L2 ↔ L1 ≡[T, d] L). +/2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.